Math · Trigonometry
Triangle calculator.
Fill in any three of the six values (at least one side) and leave the rest blank. Sides a, b, c sit opposite angles A, B, C, in degrees.
Sides a = 10 and b = 12 with angle A = 30° (the SSA case) fit two different triangles: one with B = 36.87°, C = 113.13°, c = 18.392 and area 55.177, and one with B = 143.13°, C = 6.87°, c = 2.392 and area 7.177.
Your numbers
Your result
Fill the form and press Calculate.
Which case do you have?
Three known values fall into one of five cases. SSS (three sides) uses the law of cosines for the angles. SAS (two sides and the angle between them) uses the law of cosines for the third side. ASA and AAS (two angles and a side) find the third angle from A + B + C = 180° and then use the law of sines. SSA (two sides and an angle opposite one of them) is the ambiguous case: it can give no triangle, one, or two.
The area comes from Heron's formula, area = √(s(s − a)(s − b)(s − c)), where s = (a + b + c) / 2 is the semiperimeter. Three angles alone only give the shape, not the size, so at least one side is required.
Worked example
SSS with a = 3, b = 4, c = 5: cos A = (16 + 25 − 9) / 40 = 0.8, so A = 36.87°, B = 53.13° and C = 90°; the semiperimeter is 6 and the area is √(6 × 3 × 2 × 1) = 6. SAS with a = 5, b = 7 and C = 49°: c² = 25 + 49 − 70 cos 49° = 28.08, so c = 5.299, A = 45.412°, B = 85.588° and the area is 13.207.
Limits
Angles are in degrees. Results are rounded for display to four decimal places. In the SSA case the calculator reports both triangles when two exist; check your problem for which one it intends.
Sources, opened 15 October 2026: OpenStax, Algebra and Trigonometry 2e, 10.1 Non-right Triangles: Law of Sines (law of sines; SSA ambiguous case with one, two or no solutions); OpenStax, Algebra and Trigonometry 2e, 10.2 Non-right Triangles: Law of Cosines (law of cosines; Heron's formula for area); Wikipedia, Heron's formula (semiperimeter s = (a + b + c) / 2).
Frequently asked questions
How do I solve a triangle with three sides?
Use the law of cosines: cos A = (b² + c² − a²) / (2bc), and the same pattern for B, then C = 180° − A − B. For sides 3, 4 and 5 this gives 36.87°, 53.13° and 90°.
What is the ambiguous case of the law of sines?
With two sides and an angle opposite one of them (SSA) the side opposite the angle may be too short to reach (no triangle), reach exactly once (one triangle), or swing to two positions (two triangles). With a = 10, b = 12 and A = 30° there are two triangles.
How is the triangle area calculated?
The calculator uses Heron's formula, area = √(s(s − a)(s − b)(s − c)) with s = (a + b + c) / 2, so it needs only the three sides once they are solved.
Why does it say I need exactly three values?
Three independent measurements fix a triangle. Entering fewer leaves it undetermined, and entering more can contradict each other. Three angles with no side only fix the shape.
Is this the same as the Pythagorean theorem calculator?
No. The Pythagorean calculator only handles right triangles. This one solves any triangle using the law of sines and the law of cosines.
Learning aid
Educational calculator with worked steps. Check answers and the required method against your own course or textbook. Nothing you type leaves your browser.